Small Sets containing any Pattern

Abstract

Given any dimension function $h$, we construct a perfect set $E \subseteq \mathbb{R}$ of zero $h$-Hausdorff measure, that contains any finite polynomial pattern. This is achieved as a special case of a more general construction in which we have a family of functions $\mathcal{F}$ that satisfy certain conditions and we construct a perfect set $E$ in $\mathbb{R}^N$, of $h$-Hausdorff measure zero, such that for any finite set $\{ f_1,\ldots,f_n\}\subseteq \mathcal{F}$, $E$ satisfies that $\bigcap_{i=1}^n f^{-1}_i(E)\neq\emptyset$. We also obtain an analogous result for the images of functions. Additionally we prove some related results for countable (not necessarily finite) intersections, obtaining, instead of a perfect set, an $\mathcal{F}_{\sigma}$ set without isolated points.